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Continuous random variables Xand have a joint probability density functionf(x)-2.0-1.2x-0.8yfor (x) in the first quadrant unit square [0,1]x[0,1] show that flxv) is...

Question

Continuous random variables Xand have a joint probability density functionf(x)-2.0-1.2x-0.8yfor (x) in the first quadrant unit square [0,1]x[0,1] show that flxv) is a legitimate probability density function on the unit square b) find the marginal densities fx(x) and fyly) compute E[X] and var(X) set up only (you needn't evaluate) integral expressions for the following probabilities P(X+Y>1) (sketch the region in the unit square) P(Y<X) (sketch the region in the unit square P(X<0.25

Continuous random variables Xand have a joint probability density function f(x)-2.0-1.2x-0.8y for (x) in the first quadrant unit square [0,1]x[0,1] show that flxv) is a legitimate probability density function on the unit square b) find the marginal densities fx(x) and fyly) compute E[X] and var(X) set up only (you needn't evaluate) integral expressions for the following probabilities P(X+Y>1) (sketch the region in the unit square) P(Y<X) (sketch the region in the unit square P(X<0.25 Y<0.25) (sketch the "numerator" in the unit square)



Answers

Suppose that $a$ continuous random variable has a joint probability density function given by $$f(x, y)=x^{2}+\frac{1}{3} x y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$. Find $$\int_{0}^{2} \int_{0}^{1} f(x, y) d x d y$$.

To evaluate this double on girl here. We're going to start with the inner integral, and this is with respect to X. So we're gonna go ahead and leave. Why? As a constant so anti driving X squared X cubed over three plus leave the constants one third. And why? And then the anti derivative of X is X squared over two. And then we can fix that. Bring that up in just a minute from 0 to 1 half eso We wanna simplify all of this, which is what's in the brackets first, and then we'll move out to the other in a girl with respect to why? From 1 to 2 and so now substituting here, uh, 0.5 year olds will do this in blue. Well, first off, if we substitute zero, it's going toe. Knock everything out. So we just need to substitute 0.5 in. And so one half. Think of it as a fraction one half cube would be 818 times three. So there'll be 1 24 and then plus Okay, one half here would be, uh, 1/4 times to would be 18 times three. That would also be 1 24. And why? So the wild stays and so now we've and then minus zero. So we've simplified all of that. So then now we're gonna anti derive from 1 to 2 with respect to why So our next step, we just go ahead and keep it. Keep going. Right here is one 24th. Why anti derivative of a constant and then plus one 24th anti derivative of why would be y squared. So let's change that toe. Why? And then the white squared, I should say, divided by two. So this is actually going to become a 40 ft down there. And then all of that is going to be evaluated. Let's switch back to green from 1 to 2. So let's see what this gets us first off to would be, um, to over 24 for just 1 12 plus two squared, um, for over 48 would also be 1 12 and then all of that subtract one 24th and then minus 1 48. So then we could just come up common denominator and the first part so we'll continue up here. So the 1 12 1 12 that would be, um, 2/12 but then multiplied by 48 so that the eight over 48 uh, and then minus to over 22/48 is the same thing. That 1/24 and then minus hope. Sorry. That should be a plus there. But eventually it turns into a minus because of the distributing the minus there and then minus another 1/48. So then we've got, um eight minus 26 minus 15 over 48. So this is our simplified for action here, and we can't simplify it any further.

You know there's probably been given the following probability distribution and I would like to find the marginal distributions not to find the marginal distribution for X. We need to integrate out the winds were entering out from 0 to 1. X plus Y. The wife. And so this gives us X. Y plus one half. Why squared evaluated from Y is 0- one. Mhm. And this is X plus one half. That's our marginal distribution threat. This acts plus 1/2 What? That's going from zero 20. Now for marginal distribution for what I notice that this is just going to be the exact same thing because you do the exact same mineral. So similarly we have F. Of why is why what? A half L&B. We want to find the probability that X is bigger than .25 and yeah why is greater than a half? Mhm. So here we need to integrate X. goes from 0 to 0.25. And why will then go from I'm sorry, extras from 0.25 to one. We have an upper amount of one. So it's 0.25-1. And why does from .5 to 1 about plus? Why the why? I'm sorry. Dx and Dy so integrating with respect to act first This is the integral from 0.5 to one of one half. Have squared plus. That's why evaluated from x 0.25- one. Do you want? Yeah. Uh huh. Yeah. So here's what we're going to do is plug in One party in 0.25 frets. That's attractive. And so this will have the interval from 0.5 to one Of .75. Y Plus .46875. Do you want? Now we integrate this and so this gives us, yeah, whenever we integrate .75 y plus .46875, we end up with 0.375 Y squared Plus .46875 Y, Evaluated from Y is 0.5- one. So then we plug in .5, we plug in one voice attractive. This is .5156- five.

Yeah that's probably been given a falling joint distribution function. I would like to begin by finding the marginal distributions triple X and Y. And the marginal effects just means to integrate out the why? As we integrate from 0-1, three X -Y. over 11. The why? And so this becomes 1/11 times three X y -1/2 White Square. His wife goes from 0 to 1. We put in one for why this is 1/11 Times three X -1/2. So this is our marginal distribution for X. And then for why are marginal for why? This means integrate out the Y about the ex. And so extras from 1-3. So we integrate from 1-3. The three X minus Y over 11. The yes. So this is 1/11 times 3/2 x. where minus Y X. Evaluated from X. is 123. So this is 1/11 times 3/2 times three square minus three Y -1/11 Times three. Have sometimes once where -Y. Times one. And when we combine like terms and simplify this tells us that are marginal distribution of why there's no you have to Y plus 12 all over 11. So that's our marginal for a while now would be we would like to know if X and y are independent and B is a simple no. And that's because we can clearly see that fxfx doesn't I'm sorry, that expects mhm times F. Y of Y does not equal our joint F of X. Y. So for that reason we know they're not independent on C. We want to find the probability that X is greater than two. Now here, all we need to focus on is the marginal for acts And so it's 1 11th Times three X -1/2. And so we just need to integrate this From 2 to 3. Remember X has an upper limit on 3? We don't need to look at the joint, we just need to look at the X. And so we integrated here. He was this 1 11th Times 3/2 expired minus one half X evaluated for Mexico's too just three. So now we're just going to plug in three Party into and then subtract. I don't know if we do. This gives us a value of seven over 11.

Yeah. It's probably want to find the marginal distributions given the joint probability distribution here. Now in order to find a marginal distributions, we need to integrate out the other herb. So F X with X. It means we integrate out the Y. That's we integrate from 0 to 1 three x minus Y over 11. Do you want? And so this gives us 1/11 times three X y minus one half. Y squared evaluated from y is zero to one point in 1.10 for Y. And then some track this gives us 1/11 times three x minus one half is our alphabet. Okay. Now, similarly for F some Y of Y we integrate out correct. And so this is 1/11 times three halves that squared minus X. Y. Evaluated from articles 123 Yeah. And so now for actually going to plug in one, we're going to plug in three and then we're going to subtract them. And whenever we do this gives us 1/11 times 12 minus two. Y besides our marginal distribution. For why? Mhm. No. And be we want to discuss that these are independent but clearly we can see that our joint distribution is not equal to the product of our marginal distributions. In order for them to be independent, the joint distribution must be equal to the product of the marginal distribution. And so since this is not true, they are not independent. No. And see we want to find the probability that X is greater than two. And so this is just the integral from 2 to 3 of just our X distribution. We do not need the Y. We just need the X distribution. And so we'll integrate from 2 to 3. This marginal that we just found, which is 1/11 times three X minus a half. So our anti derivative here, it will be 1/11 times three has X squared minus one half X evaluated from access to 23 And so then now we're going to plug into 40 in three and then some traveling. And or if we evaluate this sonia was seven overall up. Yeah.


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